Why We Misjudge Size: The Psychology of Seeing How Big Things Are

The eye is a superb judge of size at arm's length and a poor one almost everywhere else. Put a familiar object in your hand and you can say how big it is within a few percent. Ask how big a whale is beside a bus, or a country beside a lake, and the same brain that handles your coffee cup misses by a factor of two, and it misses in the same direction as nearly everyone else.

That is what makes Scale of Things both fun and humbling. The reasons are well studied, and knowing them is the fastest way to stop falling for them.

What reaches the back of the eye is an image whose size depends on two things at once: how big the object is and how far away it is. A coin held close and a building far off can cast exactly the same image. The Moon, a body 3,475 km across, covers about half a degree of sky, so little that a pea held at arm's length will hide it.

So the brain never sees size directly. It infers it, by combining the size of the image with an estimate of distance. When the distance estimate is good, the result is excellent. When it is wrong, or missing, so is the size.

The brain's correction for distance is called size constancy. A friend walking away from you does not seem to shrink, though their image on your eye halves every time their distance doubles. The correction is so automatic that you cannot switch it off.

Its limits show when distance cues are taken away. In a classic 1941 study, Alfred Holway and Edwin Boring had people compare discs set at different distances down a long corridor. With both eyes and a normal view, people judged the discs' real sizes well. As the cues were stripped away, one eye, then a peephole, then a darkened tunnel, their judgements slid toward the size of the image on the eye, and the far discs looked smaller and smaller.

The correction can also be fooled. An afterimage, the spot left after looking at a bright light, looks larger when you look at a far wall than at your hand, because the brain assumes it lies on whatever surface you are looking at. That is Emmert's law, from 1881. The Ames room, a distorted room built so that one corner is much further away than it looks, uses the same assumption to make a person in one corner look like a giant and a person in the other like a child.

The best-known size illusion is overhead every month. A full Moon rising over the horizon looks huge, and the same Moon high in the sky a few hours later looks ordinary. A camera shows they are the same size; if anything, the rising Moon is very slightly smaller in the image, because it is further away by about the radius of the Earth.

The Moon illusion was discussed by Ptolemy and by Ibn al-Haytham, and it still has no single agreed explanation. The leading account, tested by Lloyd Kaufman and Irvin Rock in 1962, is that we see the sky as a flattened dome, so the horizon seems further away than the zenith. Size constancy then does its job on a Moon it believes is further off, and enlarges it. Whatever the full answer, the lesson is that a single wrong assumption about distance can change how big something looks by a large margin.

Even with no distance involved at all, the things around an object change how big it looks.

In the Ebbinghaus illusion, two identical circles are drawn, one surrounded by large circles and one by small ones. The one among small circles looks clearly bigger. The figure is named for Hermann Ebbinghaus and was made famous by Edward Titchener in 1901, and it has been studied ever since because the effect is so strong and so stubborn: knowing the circles are equal does not make them look equal.

The Ponzo illusion, described by Mario Ponzo in 1911, draws two equal bars between two converging lines, like railway tracks running into the distance. The bar nearer the point where the lines meet looks longer. The lines suggest depth, the brain decides that bar is further away, and size constancy enlarges it.

A related effect catches people with monuments. Vertical lines look longer than horizontal lines of the same length, which is why almost everyone sees the Gateway Arch in St Louis as taller than it is wide, though both are 192 metres. Our piece on how big famous monuments really are has more landmarks that look a different size from the one they are.

On a board with two shapes, the labelled one is both a ruler and a trap. In 1974 Amos Tversky and Daniel Kahneman showed how strongly a number you have just seen drags an estimate toward it. People spun a wheel that stopped at 10 or at 65, then estimated the percentage of African countries in the United Nations. Those who saw 10 gave a median guess of 25; those who saw 65 gave 45. The wheel had nothing to do with the question. They called the effect anchoring.

A reference on the board is an anchor you can see, and in Scale of Things the target starts at about the reference's size, which makes the pull physical as well as mental. Add Harry Hollingworth's finding from 1910, that judgements of a quantity drift toward the middle of the range a person has in mind, and the pattern is predictable: guesses huddle near the reference. Things much bigger than it are guessed too small, and things much smaller are guessed too big.

There is a useful side to familiar size, too. In 1951 William Ittelson showed people playing cards of unusual sizes in a setting with few other cues, and they judged an oversized card to be nearer, not bigger: the brain trusted what it knew about playing cards over the image. In the game that is exactly why a well-chosen reference helps. You know how big a school bus is, so its silhouette gives every other shape on the board a scale.

Lengths are judged fairly accurately. Areas are not. In S. S. Stevens's measurements of perception, apparent area grows more slowly than real area, roughly as its 0.7th power, and Robert Teghtsoonian confirmed the effect for drawn figures in 1965. A shape with twice the area looks only about 60% bigger. A shape with ten times the area looks about five times bigger.

That is one reason big countries and lakes are so often guessed too small, and why a wall map that stretches the north, as the Mercator projection does, makes things worse. Our piece on how big countries really are shows how far off the familiar map is.

The last trap is not in the eye at all, but in how we hold numbers. Asked to place numbers on a line from 0 to 1,000, young children squash the large ones together and spread the small ones apart, as if the line were logarithmic; Robert Siegler and Julie Opfer showed in 2003 that this gives way to an even, linear placement as children get older. With very large numbers, adults do the same thing. David Landy, Noah Silbert and Aleah Goldin found in 2013 that many adults put one million near the middle of a line running from one thousand to one billion, where it belongs barely a tenth of a percent of the way along.

So the words "thousand", "million" and "billion" feel like neighbouring steps when each is a thousand times the last. When sizes range from a louse to a star, as they do in the game, it pays to think in factors of ten on purpose rather than trust the feeling of a number.

Scale of Things removes some of these traps and keeps others on purpose. Both silhouettes stand on one ground line, side by side, drawn to one scale, so there is no distance to misjudge and size constancy has nothing to correct. What remains are the effects of context, anchoring, the pull to the middle, the compression of area and the compression of big numbers. Each has a counter:

1. Decide how many times bigger or smaller before you touch the target, then make your first move a big one.
2. Count copies of the reference along the target rather than judging the target on its own.
3. Lean past your instinct when the gap is large: big things deserve a bigger guess, and small things a smaller one.
4. For areas, judge the width, not the whole shape.
5. At each reveal, look at which side of the true shape your see-through guess lands on. That is your personal direction of error.

Our guide on how to get better at Scale of Things turns those into habits. The rest is practice: play a few runs and watch the misses lean, then lean back.